一种使用GenEO谱粗空间的极限学习机全对偶区域分解方法
A Fully Dual Domain Decomposition Method for Extreme Learning Machines Using GenEO Spectral Coarse Spaces
- KAIST（韩国科学技术院）
- Graduate School of AI for Math, KAIST（韩国科学技术院人工智能数学研究生院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种用于极限学习机的新型非重叠区域分解方法，通过GenEO粗空间和加性Schwarz预处理器加速，在Stokes和波动方程上优于现有方法，并实现4096个子域的弱可扩展性。
AI中文摘要：
极限学习机（ELM）是一类机器学习算法，它随机初始化并随后固定隐藏层参数，并使用最小二乘法求解最后一层的系数。ELM已被成功应用于求解各种偏微分方程，但求解大规模最小二乘问题的成本限制了其在大型问题或需要高精度时的使用。为解决这一限制，研究者已为ELM开发了区域分解方法（DDM），通过并行计算减少训练时间。然而，这些方法，即使包含粗空间的方法，在可扩展性方面仍面临困难。本文提出了一种新的非重叠DDM用于ELM，该方法同等对待并强制执行连续性和通量条件。构建了加性Schwarz预处理器和GenEO（重叠区域中的广义特征值问题）粗空间以加速该方法。数值实验表明，新方法在迭代次数和精度方面均优于先前的ELM区域分解方法，特别是对于Stokes方程和波动方程。此外，当采用GenEO粗空间时，在多达4096个子域中观察到了弱可扩展性。
英文摘要：
Extreme learning machines (ELMs) are a class of machine learning algorithms that randomly intialize and subsequently fix the hidden layer parameters, and solve for the last layer coefficients using a least squares method. ELMs have been successfully applied to the solution of various partial differential equations, but the cost of solving large least squares problems limits their use in large scale problems or when high accuracy is desired. To address this limitation, domain decomposition methods (DDMs) have been developed for ELMs, reducing training times via parallel computation. Yet these methods, even those incorporating a coarse space, have struggled with scalability. This paper introduces a novel nonoverlapping DDM for ELMs that treats and enforces continuity and flux conditions equally. Additive Schwarz preconditioners and GenEO (Generalized Eigenvalues problems in the Overlaps) coarse spaces are constructed to accelerate the method. Numerical experiments show the new method outperforms previous DDMs for ELMs in both iteration count and accuracy, especially for the Stokes and wave equations. Furthermore, weak scalability has been observed in up to 4,096 subdomains when employing GenEO coarse spaces.
